The given equation in the problem is:

$|z + 3i| = 2$

Explanation:

1. Interpreting the equation:

   • Here, $z$ is a complex number, $z = x + yi$, where $x$ is the real part and $y$ is the imaginary part.
   • The modulus of $z + 3i$ represents the distance of the complex number $z$ from the point $-3i$ in the complex plane.

2. Geometric meaning:

   • The equation $|z + 3i| = 2$ means that the set of all points $z$ in the complex plane is at a fixed distance of 2 units from the point $-3i$.
   • This represents a circle in the complex plane.

3. Equation of the circle:

   • The center of the circle is $(0, -3)$ (i.e., $x = 0$, $y = -3$).
   • The radius of the circle is $2$.

Graph:

The circle is centered at $(0, -3)$ with a radius of $2$. To sketch it:

• Plot the center of the circle at $(0, -3)$.
• Draw a circle with a radius of $2$, extending from $(0, -5)$ to $(0, -1)$ vertically and from $(-2, -3)$ to $(2, -3)$ horizontally.

Let me know if you'd like me to generate a graph or provide more details.

Here are 5 follow-up questions to expand on this topic:

1. How would the graph change if the equation was $|z + 3i| = 3$?
2. What is the general equation for a circle in the complex plane?
3. How does the modulus $|z|$ relate to distances in the complex plane?
4. Can you describe the equation geometrically if $z$ was shifted horizontally instead?
5. How does this concept apply to real-world applications involving complex numbers?

Tip: Always think of the modulus $|z - a| = r$ as a circle, where $a$ is the center and $r$